arrow_back Prove that if \(X\) is a random variable with mean \(E X\) and \(c \in\) is \(a\) real number, then \(\operatorname{Var}(X)=E\left(X^{2}\right)-E(X)\)

Question

Prove that if \(X\) is a random variable with mean \(E X\) and \(c \in\) is \(a\) real number, then \(\operatorname{Var}(X)=E\left(X^{2}\right)-E(X)\)

Answer 1

Expanding out the square in the definition of variance gives
\[
\begin{aligned}
\operatorname{Var}(X) &=E\left[(X-E X)^{2}\right] \\
&=E\left[X^{2}-2 X E X+(E X)^{2}\right] \\
&=E\left[X^{2}\right]-E(2 X E X)+E\left((E X)^{2}\right) \\
&=E\left[X^{2}\right]-2 E X E X+(E X)^{2} \\
&=E\left[X^{2}\right]-(E X)^{2}
\end{aligned}
\]
where the third equality comes from linearity of \(E\) (Exercise \(2.3\)
(a)) and the fourth equality comes from Exercise \(2.3(\mathrm{~b})\) and the fact that since \(E X\) and \((E X)^{2}\) are constants, their expectations are just \(E X\) and \((E X)^{2}\) respectively.